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\author{UID \underline{\hspace{4cm}} \hspace{1cm} NAME \underline{\hspace{4cm}} }
\title{Mathematical Writing Exercise Chapter 02 (2.5-2.9)}
%\date{\vspace{-3ex}}
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%\date{2023 年 10 月 31 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Consider the following extract. 

\begin{center}
\fbox{
\begin{minipage}{12cm}
Let $\hat{H_k} = Q_k^H \tilde{H_k}Q_k$, partition $X = [X_1 X_2]$ and let $\mathcal{X} = \text{range} (X_1)$. 
Let $U^*$ denote the nearest orthonormal matrix to $X_1$ in the 2-norm.
\end{minipage}
}
\end{center}
%\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  These two sentences are full of potentially confusing notation. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The distinction between the hat and the tilde in $\hat{H_k}$ and $\tilde{H_k}$ is slight enough to make these symbols difficult to distinguish. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The symbols $\mathcal{X}$ and $X$ are also too similar for easy recognition. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It would be more consistent to give $\mathcal{X}$ a subscript 2. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The name $H_k$ is unfortunate, because $H$ is being used to denote the conjugate transpose, and it might be necessary to refer to $\tilde{H_k}^H$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Since $A^*$ is a standard synonym for $A^H$, the use of a superscripted asterisk to denote optimality is confusing.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 02
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  As this example shows, the choice of notation deserves careful thought. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Good notation strikes a balance among the possibly conflicting aims of being readable, natural, conventional, concise, logical, and aesthetically pleasing. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  As with definitions, the amount of notation should be maximized. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Although there are twenty-six letters in the alphabet and nearly as many again in the Greek alphabet, our choice diminishes rapidly when we consider existing connotations. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Traditionally, $\epsilon$ and $\delta$ denote small quantities, $i, j, k, m$, and $n$ are integers (or $i$ or $j$ the imaginary unit), $\lambda$ is an eigenvalue, and $\pi$ and $e$ are fundamental constants; $\pi$ is also used to denote a permutation. These conventions should be respected. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  But by modifying and combining eligible letters we widen our choice. Thus $\gamma$ and $A$ yield, for example, 
$\hat{A}$, $\bar{A}$, $\tilde{A}$, $A'$ , $\gamma_A$, $A_\gamma$, $\mathrm{A}$, $\mathit{A}$, $\mathcal{A}$, $\mathbb{A}$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Particular areas of mathematics have their own notational conventions. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  For example, in numerical linear algebra lowercase Greek letters represent scalars, lowercase roman letters represent column vectors, and uppercase Greek or roman letters represent matrices. This convention was introduced by Alston Householder.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 03
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Mathematicians are always searching for better notation. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Knuth describes two notations that he and his students have been using for many years and that he thinks deserve widespread adoption. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  One is notation for the Stirling numbers. The other is the notation $[\mathcal{S}]$, where $\mathcal{S}$ is any true-or-false statement. The definition is
\begin{eqnarray*}
[\mathcal{S}] = \left\{ \begin{array}{ll}
1, & \mathrm{if }\,\, \mathcal{S} \,\, \mathrm{is\,\, true}, \\ 
0, & \mathrm{if}\,\, \mathcal{S} \,\,\mathrm{ is\,\, false}.
\end{array}\right.
\end{eqnarray*}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  For example, the Kronecker delta
\begin{eqnarray*}
\delta_{ij} = \left\{ \begin{array}{ll}
1, &i=j, \\ 
0, &i\neq j,
\end{array}\right.
\end{eqnarray*}
can be expressed as $\delta_{ij} =[i =j]$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The square bracket notation will seem natural to those who program; indeed, Knuth adapted it from a similar notation in the 1962 book by Kenneth Iverson that led to the programming language APL. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The square bracket notation is used in the textbook Concrete Mathematics; that book and Knuth's paper give a
convincing demonstration of the usefulness of the notation. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 04
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Paul Halmos has these words to say about two of his contributions to mathematical notation. 
... \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  My most nearly immortal contributions to mathematics are an abbreviation and a typographical symbol. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  I invented ``iff'', for ``if and only if'' -- but I could never believe that I was really its first inventor. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The symbol is definitely not my invention -- it appeared in popular magazines (not mathematical ones) before
I adopted it, but, once again, I seem to have introduced it into mathematics. 
... \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It is the symbol that sometimes looks like $\square$, and is used to indicate an end, usually the end of a proof. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It is most frequently called the ``tombstone'', but at least one generous author referred to it as the ``halmos''.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 05
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  In formal writing mathematics is written in complete sentences with proper punctuation, so every piece of mathematics should belong to a sentence. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Hence mathematical expressions should always be punctuated. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  In the following display, all the punctuation marks are necessary. 
The second displayed equation could be moved in-line in order to put less emphasis on the definition of vector $p$-norm. 

\begin{center}
\fbox{
\begin{minipage}{12cm}
The three most commonly used matrix norms in numerical analysis are particular cases of the H$\ddot{o}$lder $p$-norm
\begin{eqnarray*}
\lVert A \rVert_p = \underset{x\neq 0}{\max} \frac{\lVert Ax \rVert}{\lVert x \rVert}, A \in \mathbb{R}^{m\times n}, 
\end{eqnarray*}
where $p > 1$ and
\begin{eqnarray*}
\lVert x \rVert_p = \left( \sum\limits_{i=1}^{n} |x_i|^p \right)^{1/p}. 
\end{eqnarray*}
\end{minipage}
}
\end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 06
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Students often make a mistake illustrated by this example of the derivation for the solution of a quadratic equation:

\begin{center}
\fbox{
\begin{minipage}{12cm}
We have
\begin{eqnarray*}
x2 + bx - c &=& 0 \\ 
(x + b/2)^2 &=& c + b^2/4 \\ 
x + b/2 &=& \sqrt{c + b^2/4} \\ 
x &=& -b/2 + \sqrt{c + b^2/4}
\end{eqnarray*}
\end{minipage}
}
\end{center}
%\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Here, a sequence of unpunctuated equations is presented and the reader is expected to make the connections between them. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Instead, a logical sequence of linked equalities should be given that make one or more sentences. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  This example can be rewritten as follows. 

\begin{center}
\fbox{
\begin{minipage}{12cm}
We have
\begin{eqnarray*}
0 = x^2 + bx - c =  (x + b/2)^2 - c - b^2/4,
\end{eqnarray*}
so that, on rearranging and taking the positive square root,
\begin{eqnarray*}
x = -b/2 + \sqrt{c - b^2/4}.
\end{eqnarray*}
\end{minipage}
}
\end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The ``We have'' that begins the previous sentence serves to introduce the following equation. 
... \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It is good style to start a sentence with an equation. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  By contrast, the sentence ``Then a = 6'' (for example) does not need ``we have'' after ``then''; it is a perfectly good sentence with subject, verb, and object. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 07
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Mathematicians are supposed to like numbers and symbols, but I think many of us prefer words. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  If we had to choose between reading a paper dominated by symbols and one dominated by words then, all other things being equal, most of us would choose the wordy paper, because we would expect it to be easier to understand. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  One of the decisions constantly facing the mathematical writer is how to express ideas: in symbols, in words, or in both. I suggest some guidelines.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Use symbols if the idea would be too cumbersome to express in words, or if it is important to make a precise mathematical statement. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Use words as long as they do not take up much more space than the corresponding symbols. 
... \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Explain in words what the symbols mean if you think the reader might have difficulty grasping the meaning or essential feature.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 08
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Example 5. 

\begin{center}
\fbox{
\begin{minipage}{12cm}
If $y_1, y_2, \cdots, y_n$ are all $\neq 1$ then $g(y_1, y_2, \cdots, y_n) >0$.
\end{minipage}
}
\end{center}
%\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  In the first sentence ``all $\neq  1$'' is a clumsy juxtaposition of word and equation and most writers would express the statement differently. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Possibilities include

\begin{center}
\fbox{
\begin{minipage}{12cm}
If $y_i\neq 1$ for $i =1,2,\cdots,n$, then $g(y_1,y_2,\cdots,y_n) > 0$. \\
If none of the $y_i (i = 1,2,\cdots, n)$ equals 1, then $g(y_1,y_2, \cdots, y_n) > 0$.
\end{minipage}
}
\end{center}
%\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  If the condition were ``$\neq 0$'' instead of ``$\neq 1$'', then it could simply be replaced by the word ``nonzero''.  
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  In cases such as this, the choice between words and symbols in the text (as opposed to in displayed equations) is a matter of taste; good taste is acquired by reading a lot of well-written mathematics.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 09
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  An equation is displayed when it needs to be numbered, when it would be hard to read if placed in-line, or when it merits special attention, perhaps because it contains the first occurrence of an important variable. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The following extract gives an illustration of what and what not to display. 

\begin{center}
\fbox{
\begin{minipage}{12cm}
Because $\delta(\bar{x},\mu)$ is the smallest value of $\lVert \bar{X}z/\mu-e \rVert$ for all vectors $y$ and $z$ satisfying $A^Ty + z = c$, we have $$\delta(\bar{x},\mu)\le \lVert \frac{1}{\mu}\bar{X}z - e \rVert. $$

Using the relations $z = \mu X^{-1}s$ and $\bar{x}_i = 2x_i - x_is_i$ gives
$$ \frac{1}{\mu}\bar{X}z = \bar{X}X^{-1}s = (2X - XS)X^{-1}s = 2s - S^2e.$$

Therefore, $\delta(\bar{x},\mu) \le \lVert 2s - S^2e - e \rVert$, which means that
\begin{eqnarray*}
\delta(\bar{x},\mu) &\le& \sum\limits_{i=1}^{n} (2s_i-s_i^2-1)^2 = \sum\limits_{i=1}^{n} (s_i-1)^4 \\ 
&\le&  \left( \sum\limits_{i=1}^{n} (s_i-1)^2 \right)^2 = \delta(x,\mu)^4. 
\end{eqnarray*}

The condition $\delta(x,\mu) < 1$ thus ensures that the Newton iterates $\bar{x}$ converge quadratically. 
\end{minipage}
}
\end{center}

\item  The second and third displayed equations are too complicated to put in-line. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The first $\delta(\bar{x},\mu)$ inequality is displayed because it is used in conjunction with the second display and it is helpful to the reader to display both these steps of the argument. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The consequent inequality $\delta(\bar{x},\mu) \le \lVert 2s - S^2e - e \rVert$ fits nicely in-line, and since it is used immediately it is not necessary to display it.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 10
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  A general principle is to avoid line breaks within fences (parentheses and brackets). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A displayed sequence of equations or inequalities should be broken before a relation symbol and aligned on a relation symbol. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  As an example,
\begin{eqnarray*}
|u_{kk}^{-1}| &=& |e_k^T U^{-1}e_k| = |e_k^T A^{-1}P^TLe_k| = |(PA^{-T}e_k)^TLe_k| \\ 
&\le & \lVert PA^{-T}e_k \rVert_1 \lVert Le_k \rVert_\infty \le \lVert A^{-1} \rVert_\infty. 
\end{eqnarray*}

\item  This example could alternatively be broken at the third equals sign or displayed over three lines. 
Where to break is largely a matter of taste.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

%When a displayed formula is too long to fit on one line it should be broken before a binary operation. Example:
%
%
%Ideally, the indentation on the second line should take the continuation expression past the beginning of the left operand of the binary operation at which the break occurred, but as this example illustrates, this is not always possible for long expressions.

\end{enumerate}

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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